Every real number has a decimal representation.

Question

I was reading this answer explaining why all real number have a decimal representation. I think it is really a nice explanation but I don't really see were (I think it is a little hidden) we use the density of $\mathbb Q$ into $\mathbb R$ (which I think somehow we should use).

Answer 1

It seems to me the linked answer is virtually a proof that $\mathbb Q$ is dense in $\mathbb R$. It shows that a subset of $\mathbb Q$—fractions with denominators which are powers of $10$—has elements arbitrarily close to any real number. Which makes the same true of $\mathbb Q$, thereby proving that $\mathbb Q$ is dense in $\mathbb R$.

So the reason it doesn't use density of $\mathbb Q$ in $\mathbb R$ is that it proves it instead.

Answer 2

It is used nowhere in that proof and I see no reason why it should.